PauliVec

                                                                                

META AI:

Qiskit PauliVec; I have a 2-qubit circuit with the first a Hadamard gate and the second a cx with q(0) as control and q(1) as target. A PauliVec representation shows the YY as negative1, while the other three are positive. What accounts for this?

A nice question about Qiskit and quantum computing!

The PauliVec representation you're referring to is likely the Pauli Vector representation of the density matrix of your 2-qubit circuit. This representation decomposes the density matrix into a linear combination of Pauli matrices.

In your case, the circuit consists of:

Hadamard gate (H) on qubit 0

Controlled-NOT gate (CX) with qubit 0 as control and qubit 1 as target

To understand why the YY term is negative, let's analyze the circuit:

Step 1: Hadamard gate on qubit 0

The Hadamard gate creates a superposition state on qubit 0: |0+ |1. This can be represented as:

ρ = |ψψ| = (|00+ |10+ |01+ |11) / 2

Step 2: Controlled-NOT gate

The CX gate applies a NOT operation to qubit 1 if qubit 0 is in the state |1. This creates entanglement between the qubits.

After applying CX, the state becomes:

ρ = |ψψ| = (|00+ |11- |01+ |10) / 2

Notice the relative phase difference between |00and |11.

Pauli Vector representation

Now, let's decompose the density matrix ρ into Pauli matrices:

ρ = (I ⊗ I + X ⊗ X + Y ⊗ Y - Z ⊗ Z) / 4

Here:

I is the identity matrix

X, Y, Z are the Pauli matrices

⊗ denotes the tensor product

The coefficients of the Pauli Vector representation are:

II: +1

XX: +1

YY: -1 (this is what you observed)

ZZ: +1

The negative coefficient for YY arises from the relative phase difference between |00and |11in the entangled state. Specifically, it's due to the antisymmetric nature of the Bell state |ψ= (|00- |11)/√2, which is present in the density matrix.

In summary, the negative YY term in the PauliVec representation is a consequence of the entanglement generated by the CX gate and the relative phase difference between the computational basis states.

Could I see a calculation of the tensor product for the two Ys?

Here's the calculation for the tensor product of two Y Pauli matrices:

Y ⊗ Y =

= |0⊗0 |0⊗1 |1⊗0 |1⊗1

|0 -i 0 0 | |0 0 -i 0 |

|0 i 0 0 | |0 0 i 0 |

|0 0 -i 0 | |0 -i 0 0 |

|0 0 i 0 | |0 0 0 -i |

Now, let's calculate the expectation value of Y ⊗ Y in the state:

ρ = (|00+ |11- |01+ |10) / 2

We'll compute the trace of the product ρ(Y ⊗ Y):

Tr[ρ(Y ⊗ Y)] =

= (∑ij ρij (Y ⊗ Y)ji) / 2

= [ρ00 (Y ⊗ Y)00 + ρ11 (Y ⊗ Y)11 + ρ01 (Y ⊗ Y)10 + ρ10 (Y ⊗ Y)01] / 2

Using ρij values from ρ:

ρ00 = ρ11 = 1/2, ρ01 = ρ10 = 0

Tr[ρ(Y ⊗ Y)] =

= [(1/2)(-i) + (1/2)(i)] / 2

= -1/2 - 1/2

= -1

So, the coefficient of YY in the PauliVec representation is indeed -1

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